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Foliations, the Ergodic Theorem and Brownian Motion View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS 51, 285-311 (1983) Foliations, The Ergodic Theorem and Brownian Motion LUCY GARNETT Baruch College, New York, New York JO010 Communicated by the Editors Received October 1982; revised December 1982 1. INTRODUCTION We combine ideas from three areas: (1) foliations, (2) the ergodic theorem, (3) Brownian motion. Regarding the leaves of a compact foliated Riemannian manifold as the orbits of a dynamical system, we will prove a Birkhoff ergodic theorem asserting that the spatial average of a function equals its time average along random paths in the leaves. There are several unexpected developments in the foundations. First, there always are measures on any compact foliated Riemannian manifold (M,F) with respect to which an adequate (brownian motion along the leaves) ergodic theorem holds. Second, relative to these measures,which are called harmonic, there is a Liouville-type theorem for any bounded Bore1 function on the manifold which is harmonic on each leaf. Namely, for almost all leaves relative to any one of these harmonic measures,this leaf harmonic function is constant on each leaf. Third, the harmonic measureshave a very reasonable local characterization. In any foliation coordinate system, they are transversal sums of h(L) dx(L), where h(L) is a positive harmonic function on the leaf L and dx(L) is the Riemannian measure on L. These statements comprise Theorem 1 which is stated more precisely later in the Introduction. The first fact is surprising because up to now the only measures attached to foliations were the holonomy invariant transverse measuresof Plante [ 141 and Ruelle-Sullivan [ 151 and some foliations do not admit these. The second fact is unexpected becausethere are foliations where all the leaves are hyper- bolic planes and each hyperbolic plane, being conformal to the disk, admits 285 0022-1236183$3.00 Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved. 286 LUCY GARNETT many nonconstant bounded harmonic functions. The third fact is interesting because the harmonic measures are first defined globally as fixed points of an operator on measures given by Brownian motion diffusion along the leaves. The local picture of harmonic measuresarises by characterizing these measureswith a differential condition that the leaf Laplacian of the measure is equal to zero. A rich and important class of examples consists of the Lie group foliations F(H, G, T), where G is a Lie group, H is a closed connected subgroup of G and T is a discrete cocompact subgroup of G. The underlying manifold M of the foliation is G/T and the leaves of the foliation are the left cosets of H in G projected to G/T. In this situation G is automatically unimodular; that is, left Haar measure equals right Haar measure, but H may or may not be unimodular. However, it is a theorem of Bowen [I ] that F(G, H, T) has a holonomy invariant transverse measure if and only if H is unimodular. For example, if G = Sl(2, R) and H is the ax + b group, and T is any cocompact Fuchsian group, then H is not unimodular and F(H, G, T) has no holonomy invariant transverse measure, but it does have a harmonic measure. Indeed, answering a question asked us by Bill Thurston, we shall demonstrate later that the harmonic measure in that case is unique. (In all these F(H, G, T) examples, the Haar measure on G determines a harmonic measure relative to a right invariant metric on G/T.) Consider the case when G = S/(2, C) and H is Sl(2, R) and T is a cocompact discrete subgroup of S/(2, C). Divide by SO(2, R), acting on the left, to get a foliation whose leaves are copies of the hyperbolic plane. Our second fact (the Liouville theorem for Bore1 leaf harmonic functions) says that it is impossible to put together nonconstant bounded harmonic functions on these leaves in a way to produce a bounded Bore1 function on the whole space. Moreover, H is a semisimple Lie group and thus unimodular. Hence, the Haar measure determines a holonomy invariant transverse measure. For this foliation, Plante’s subexponential growth method (141 for constructing such measures does not apply. However, our ergodic decomposition theory, developed in Section 6, gives a way to view this measure as a limit of diffused” Dirac measures. If (M, F) is a compact foliated Riemannian manifold, we say that a probability measure m on M is harmonic if (df, m) equals zero, where f is any bounded measurable function on M which is smooth in the leaf direction and d denotes the Laplacian in the leaf direction. (We use the symbol (g, U) to denote the integral of a function g with respect to the measure u.) THEOREM 1. (a) A compact foliated Riemannian manifold (M, F) always has nontrivial harmonic measures. (b) Any bounded Bore1 function h which is harmonic on each leaf must be constant on almost all leaves, relative to any finite harmonic measure. BROWNIAN MOTION ON FOLIATION 287 (c) A measure m is harmonic if and only if m locally (in a distinguished foliation coordinate system) disintegrates into a transversal sum of leaf measures, where almost every leaf measure is a positive harmonic function times the Riemannian leaf measure. The proof of Theorem 1 is given in Section 3. We actually prove stronger versions of (b) and (c). Manifold M need not be compact, but (M, F) must satisfy a condition of bounded geometry for the leaves, defined in Section 2. One application to complete Riemannian manifolds partially generalizes a result of Yau. Theorem 1 gives basic information relevant to our Brownian motion ergodic theorems which are stated and proved in Section 5. The main conceptual point of these theorems is that Brownian motion along the leaves makes the k-dimensional leaves of the foliation behave like the orbits of an action of the semigroup of positive reals. The basic analytic estimates for the proofs involving Brownian .motion come from Malliavin [ lo] and McKean [ 121. McKean provided many useful and helpful comments throughout this work. Sullivan asked the original motivating question: Namely, what can be said about the dynamics of the leaves of a foliation; how do they wind around in a manifold; how often do they pass through a given flow box ? We give a general answer to these questions by sampling the leaf along a random path of Brownian motion. A geometric application to average Gaussian curvature of leaves combines our ergodic theorem and a foliation result of Connes, Section 5. 2. PRELIMINARIES A foliation F is a partition of a manifold M into k-dimensional connected submanifolds L which is locally trivial but usually globally complicated. Namely, M can be regarded as the union of open disks in Euclidean n space so that the overlap maps are smooth and preserve the foliation of Euclidean space by a parallel family of k planes. These coordinate disks (usually denoted E here) are called flow boxes or distinguished (foliation) coordinate systems. The quotient of such an E by the horizontal k planes is called the quotient transversal Z = Z(E). If p: E + Z is the projection, then any measure m on E may be disintegrated uniquely into the projected measure y on the transverse Z and a system of measures a(s) on the leaf slices p-‘(s) = E(s) for each s in I. These measuressatisfy the following conditions: (1) a(s) is a probability measure on E(s). (2) If S is a measurable subset of I, then y(S) = mW’(W (3) F ix any measurable subset B of E, then a(s)(B n E): I+ R is a measurable function of s. (4) If f is m integrable and supp(f) < E, then If (4 dmW = J‘f (v) dWW 4(s). Our theory necessitatesa condition of uniformly bounded geometry on the 288 LUCY GARNETT leaves of the foliated space. Namely, there is some constant K such that given any point in M, there is a coordinate map v, taking the unit ball in the leaf about that point diffeomorphically onto the unit ball in Euclidean space with the derivatives of (o up to order 3 bounded by K; in particular, the lengths of tangent vectors at the point are distorted by less than K. If the foliated space is compact, then we know that this condition is satisfied. However, there will be situations where an argument works in a more general context than compactness. In fact our proof of Theorem l(b) (c) does not require compactness, but only uses the condition of uniformly bounded geometry. This property has several consequences.Since sectional curvature is the second derivative of the metric, we have that the sectional curvature is bounded from above and below. The injectivity radius, given by injecting the tangent space into the manifold, can also be bounded below by one. TThe third consequence is that the volume of a leaf grows at most exponentially. We provide the tangent bundle to the foliation with a C3 Riemannian structure. Each leaf L of the foliation inherits a Riemannian structure making it into a connected C3 Riemannian manifold complete for diffusion. (The integral of the heat kernel over the whole space equals one.) We have a Laplace-Beltami operator A, on each leaf L. The measure on a leaf induced by the Riemannian metric is denoted by dx.
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